1,2

A configuration is a set of triples (of points) where every pair of points occurs in at most one triple. (A Steiner triple system is a set of triples where every pair occurs exactly once; thus configurations are often called partial Steiner triple systems). The triples are also called blocks.

A 'generator' is 'a configuration where every point occurs in at least two blocks'. The term refers to the work of Horak, Phillips, Wallis & Yucas, who show that the number of occurrences of a configuration in a Steiner triple system is expressible as a linear form in the numbers of occurrences of the generators.

If we relax the restriction on the number of times a pair of points can occur in a configuration - so that a configuration is just any multi-set of triples - then we get A050913.

If we allow a configuration to be any *set* of triples - i.e. configurations with multiple occurrences of blocks are not allowed, but more than one pair is allowed - then we get A058790.

Forbes, Grannell & Griggs, 'Configurations and trades in Steiner triple systems', in preparation.

Mike Grannell and Terry Griggs, 'Configurations in Steiner triple systems', in Combinatorial Designs and their Applications, Chapman & Hall, CRC Research Notes in Math. 403 (1999), 103-126.

Horak, P., Phillips, N. K. C., Wallis, W. D. and Yucas, J. L., Counting frequencies of configurations in Steiner triple systems. Ars Combin. 46 (1997), 65-75.

The five configurations of 3 triples are

*---*---* ..... *---*---* ...

*---*---* ..... *---*---* ...

*---*---* ...... \ ..........

................. * .........

.................. \ ........

.... * ..... * .... * .......

... / \ ... / ...............

.. * . * . * .............. *

. / ... \ / .............. /|

.* ..... * .............. * |

........................ /. |

*---*---*---*---* ..... * . *

........ \ ............. \ .|

......... * ............. * |

.......... \ ............. \|

........... * ............. *

Cf. A082790, A050913, A058790.

Sequence in context: A141201 A052891 A052815 this_sequence A072110 A114296 A121689

Adjacent sequences: A082786 A082787 A082788 this_sequence A082790 A082791 A082792

nonn,nice

T. Forbes (anthony.d.forbes(AT)googlemail.com), May 24 2003